Nodal solutions for a second-order m-point boundary value problem

Abstract

We study the existence of nodal solutions of the m-point boundary value problem

$$\begin{gathered} u'' + f(u) = 0,0 < t < 1, \hfill \\ u'(0) = 0,u(1) = \sum\limits_{i = 1}^{m - 2} {\alpha _i u(\eta i)} \hfill \\ \end{gathered} $$

where η _i ∈ ℚ (i = 1, 2, ..., m − 2) with 0 < η ₁ < η ₂ < ... < η _m−2 < 1, and α _i ∈ ℝ (i = 1, 2, ..., m − 2) with α _i > 0 and $\sum\nolimits_{i = 1}^{m - 2} {\alpha _i }$ < 1. We give conditions on the ratio f(s)/s at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.